Properties of the infinite-dimensional Riccati equation arising in the many-fermion problem

1996 ◽  
Vol 60 (2) ◽  
pp. 221-222
Author(s):  
G. V. Koval'
Keyword(s):  
Riccati Equation ◽  
Infinite Dimensional ◽  
The Many

COMPLETE INTEGRABILITY AND THE PAINLEVÉ PROPERTY

1990 ◽  
Vol 04 (09) ◽  
pp. 1483-1516 ◽  
Author(s):  
SANJAY PURI
Keyword(s):  
Equations Of Motion ◽  
Singularity Analysis ◽  
Complete Integrability ◽  
Tabular Form ◽  
Rigorous Results ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Painlevé Property ◽  
Hamiltonian Dynamical Systems ◽  
The Many

We present a systematic review of the connection between the complete integrability of (finite-dimensional and infinite-dimensional) Hamiltonian and non-Hamiltonian dynamical systems and the singularity structure of their equations of motion. We make a special attempt to separate the many conjectures in the field from the few rigorous results. The singularity analysis of ordinary and partial differential equations is illustrated by means of four examples. In the Appendix, we present a brief review of the special properties of soliton equations and the relations between them. We summarize these relations in a tabular form.

QUANTUM NONLOCAL FIELD THEORY: PHYSICS WITHOUT INFINITIES

1992 ◽  
Vol 07 (24) ◽  
pp. 6121-6157 ◽  
Author(s):  
N.J. CORNISH
Keyword(s):  
Field Theory ◽  
Relativistic Quantum ◽  
Physical Constraints ◽  
Infinite Dimensional ◽  
Nonlocal Field Theory ◽  
Functional Measure ◽  
Finite Theory ◽  
Nonlocal Field ◽  
The Many ◽  
Novel Applications

It is argued that nonlocality is an essential ingredient in Relativistic Quantum Field Theory in order to have finite theory without recourse to a renormalisation program. A critical review of the physical constraints on the form the nonlocality can take is presented. The conclusion of this review is that nonlocality must be restricted to interactions with the vacuum sea of virtual particles. A successful formulation of such a theory, QNFT, is applied to scalar electrodynamics and serves to illustrate how gauge invariance and manifest finiteness can be achieved. The importance of the infinite dimensional symmetry groups that occur in QNFT are discussed as an alternative to supersymmetry, the ability to generate masses by breaking the nonlocal symmetry with a noninvariant functional measure is given a critical assessment. To demonstrate some of the many novel applications QNFT may make possible two examples are mooted, the existence of electroweak monopoles and the formulation of a finite perturbative theory of Quantum Gravity.

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